On a Leslie–Gower predator–prey model incorporating a prey refuge

We propose a Leslie–Gower predator–prey model incorporating a prey refuge. By constructing a suitable Lyapunov function, we show that the unique positive equilibrium of the system is globally stable, which means that for this ecosystem, prey refuge has no influence on the persistent property of the system. Mathematic analysis shows that increasing the amount of refuge can increase prey densities. As far as the predator species is concerned, when the assumption a1r2≤a2b1a1r2≤a2b1 holds, increasing the amount of prey refuge can decrease the predator densities; when the assumption a1r2>a2b1a1r2>a2b1 holds, there exists a threshold m∗m∗, such that for the prey refuge smaller than this threshold, increasing the amount of prey refuge can increase the predator densities and if the prey refuge is larger than the threshold, increasing the amount of prey refuge can decrease the predator densities.